PHI269 Midterm

MASON KELLERWabash College

March 25, 2015

Part One

Prompt

Our consideration of views in the philosophy of mathematics has cen-tered on the following three major areas.Ontology: What are mathematical objects? What makes mathematical state-ments true?Epistemology: How do we know anything about mathematical objects?How do we know that mathematical statements are true?Applicability: How does mathematics apply to the physical world?For each of the following four philosophers (Plato, Aristotle, Kant, Frege),(a) explain how they answer these questions, and (b) articulate an objectionto the account you describe in part (a).

Plato

Plato considers mathematical objects to be abstract objects independent fromourselves. I hesitate to describe them as forms instead of objects; there is oneperfect form of the colour white, but there are many equally perfect circles.In this way, mathematical objects are beneath forms in terms of abstractness.Indeed, Plato gives us this in the analogy of the divided line. However, I alsothink that Plato would agree that a wheel instantiates the idea or form of acircle.

Now, in order to elaborate on how we know about the relationships ortheorems between these objects, one must first tackle the topic of anamnesis.As presented in the Meno, anamnesis is the theory that all mathematical rela-tionships are fixed and already existent: one recalls them from pre-incarnationexperiences with the objects themselves. Mathematical proof, that is, deductivereasoning, is merely an aid to this recollection, and a means to convince usthat we are remembering correctly. Despite this, I think the modern positionof mathematical Platonism is that we know theorems by means of deductivereasoning about the objective qualities of these intelligible objects. That said,modern views do not retroactively change what Plato himself put forward.

To answer Wigner’s question about why mathematics is unreasonably ef-fective at modeling the material world, one must consider in more detail thetheory of the forms. Consider that physical objects are imperfect reflectionsor instantiations of the intelligible objects. In this way, it makes sense thatmathematical relationships model the physical world. In Plato’s view, it would

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almost be more appropriate to say that the physical world is a model of themathematical world, and not vice-versa.

I would personally prefer it if the Platonist thesis were true; it wraps every-thing up in a nice little bow. However, the glaring objection to Platonism isthat there is no objective evidence for the existence of forms. It is not a logicalposition to take. Moreover, as a mathematician, not a philosopher or aesthete,beauty is in parsimony. To presuppose the existence another world of intelligi-ble objects, past lives, trans-incarnation recollection, and some mechanism bywhich intelligible objects are instantiated imperfectly is, to be gentle, an awfullot. If there is no means by which to come to know the forms objectively, thenPlatonism is more of a religion then not. Indeed, I liken it to Gnosticism: one canonly learn the knowledge necessary for salvation personally and miraculously.One can only claim the forms’ existence through similar epiphany.

Aristotle

Aristotle presents a more empirical thesis about mathematics. His claim isthat mathematical objects are merely physical objects, or at least direct abstrac-tions from physical objects. In this view, many questions one might ask . Amathematician would certainly desire to show that, say, "four divides twelve" isalways a true statement. But to Aristotle, this is a contingent truth: we mightconceive of a world wherein there are no groups of twelve. Then, to Aristotle, itwould be illegitimate to reason about the "universal" twelve. This is because inAristotelian logic, "four divides twelve" implies that there is "four" and there is"twelve." But in the world just conceived, there is no "twelve."

Kline tells us by way of reference to Aristotle’s Posterior Analytics that Aris-totle thought mathematical statements were true by means of deductive logicfrom axioms. However, Aristotle’s view is a posteriori in that these axioms areknown to be true by means of "our infallible intuition." In contrast to Plato,physical experience is necessary in order to intuit these principles.

If mathematical objects exist only as objects of the physical world it meansthat their relationships should readily apply to the physical world. I will makespecial contrast against Plato, since their answers sound so similar, and I findthem to be far apart in the mind. Plato wants that there are perfect objectswith perfect relationships, and imperfect reflected objects with imperfect re-lationships. Aristotle wants that there are only physical objects and physicalrelationships. Mathematics merely separates away irrelevant characteristics ofthose objects, and in this way does not lose any applicability. Indeed, I think thatAristotle would agree that if one could separate the relevant from the irrelevantperfectly, one could mathematically derive formulae that predict perfectly. Errorfor Plato is due to systemic flaw in the physical. For Aristotle, error arises fromflaws in our art.

I touched upon my central objection to this philosophy above, namely thatit is highly unsatisfying to only know contingent truths. That is to say, it isinherently less satisfying than knowing eternal truths. Moreover, the viewbreaks down when applied to the very abstract. I have in the past advocatedthe sensualist thesis: "there is not anything in mind, which hasn’t been in the

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sensations," at least in regard to the subjective (see my Intuition and Sensualismin The Wabash Commentary: Soul and the City). But it is objective that there aretopologies wherein a sequence can converge into more than one point. How canthis originate from the senses? There is no way to perceive such a space, indeed,that sequences converge to at most one point is a necessary property to allmodels that fit physicality. Is such a space not a mathematical object, because itdoes not result from a separation of qualities from an object? Is it illegitimate toposit about? Affirmative answers to these questions are incredibly dissatisfyingto the mathematician, but the Aristotelian view seems to mandate those sameanswers.

Kant

Kant’s central claim is that mathematics is a body of synthetic a priori knowl-edge. That is to say, it is a body of truths which are necessary but not containedwithin the initial suppositions. Kant views mathematical objects as a conceptu-alist. That is to say, to him, they reside in the mind. Frege describes Kant’s claimabout ontology to be psychological. This is because Kant has the position thatthe world we perceive is perceived precisely because of the human mind. Thatis to say that space, distance, and form are concepts imposed on raw sensorydata from a world that is ultimately unknowable. Moreover, Kant tells us thatthis perception is consistent across all human minds and that we cannot helpbut sense space as Euclidean. This tells us why Kant wants to believe thatmathematical truths are necessary. Kant supports his claim about the syntheticnature of mathematical truths by way of example. He gives us the example of5 + 7 = 12, and instructs us to observe that 12 is not contained within any ofthe ideas that make up 5 + 7 = 12.

The psychological thesis wraps up the epistemological questions and Wigner’squestion of effectiveness very neatly. We know our a priori geometrical truthsbecause it is not, and cannot, be conceivable to us that construction upon theobject in question changes the properties of the object in question, rather it is aproperty of the mind mode of formatting sensory data that makes the propertiesof any given triangle invariant under additional constructions. This knowledgeof geometry is then a necessary feature of space. From this it is easy to see whyempirical laws, describing space and being formulated in the language of math-ematics, are useful and accurate: the mathematical nature of these relationshipsis built into space itself.

One particular objection to Kant’s view is presented in greater detail in PartTwo of this exam. For completeness here, I present an abbreviated form. Thecentral thrust of the issue is that human psychology is more vast than Kantever assumed. Really, claims made from his so called "infallible intuition" arecontingent. Said another way, even things such as arithmetic and geometry arecontingent upon the mind of each man. Then they cannot be a priori, so the viewis inconsistent.

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Frege

Frege was a logicist. That is to say, Frege believed that all of mathematicsis reducible to logic. Then it certainly makes sense to ask instead questionsabout the ontology and epistemology of Frege’s logical system. Bostock tellsus that Frege was a realist in a Platonic way, but swiftly moves on. Frege inhis The Thought: A Logical Inquiry shows that he believes thoughts, like thePythagorean Theorem, to be distinct from ideas, which Frege characterises assense-impressions, decisions, moods, and imaginative creations. Frege thendistinguishes how his ideas are different from physical objects that belong towhat he calls the "outer world." Finally, he uses these distinctions to show thatthoughts cannot be ideas. I provide his conclusion, which shows his realistposition:

"A third realm must be recognized. What belongs to this correspondswith ideas, in that it cannot be perceived by the senses, but withthings. in that it needs no bearer to the contents of whose conscious-ness to belong. Thus the thought, for example, which we expressedin the Pythagorean theorem is timelessly true, true independently ofwhether anyone takes it to be true. It needs no bearer. It is not truefor the first time when it is discovered, but is like a planet which,already before anyone has seen it, has been in interaction with otherplanets."

Frege also holds, of course, that we know about these objects by reasoninglogically.

Frege claims in §106 his The Foundations of Arithmetic that "a number as-serts something objective of a concept." While not objects in and of themselves,concepts contain objects, and so number can at least tell us about the relation-ships between objects. This means precisely that it is reasonable to expect thatnumbers are good models for the world; a model is only a set of relationships.

My central objection to Frege is to his ontic claims. They remind me stronglyof Platonism, and in that regard some of the objection to Plato that I renderedabove apply. Another concern is that Frege’s project is to base arithmetic onlogic, but simple truths of arithmetic like 2 + 2 = 4 are, in a way, more obviousthan some of his rules of logic. Shouldn’t we, for certainty’s sake, base our artoff the most obvious principles?

Part 2

PromptMathematical developments, primarily but not exclusively in the 19th

century, complicated the philosophy of mathematics considerably. Pleasechoose one such development, and explain how it posed a problem for theviews of Plato, Aristotle, or Kant that you discussed in Part One. This partcan correctly be addressed in approximately 500 words.

Non-Euclidean geometries are geometries in which Euclid’s parallel linepostulate, or an equivalent, is negated. For ease, consider the equivalent axiom

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due to Playfair: "In a plane, given a line and a point not on it, at most oneline parallel to the given line can be drawn through the point." The two non-Euclidean geometries arise out of the negation. Supposing that there are noparallels yields the elliptic or spherical geometry. Supposing there are at leasttwo parallels gives the hyperbolic or "saddle" geometry. Certain theorems areinvariant between geometries, but there are many that are not.

The discovery of non-Euclidean geometries was disastrous to the Kantianview of mathematics. Kant tells us that Newtonian mechanics and Euclideangeometry are truths, created by our minds’ imposition of form on a reality thatis ultimately unknowable. Moreover, Kant tells us that we cannot possiblyperceive differently. But we now know that it is not possible to tell with thesenses the true geometry of the space that we perceive. However, I believethat it would be very easy to generate humans who perceive only sphericalgeometry! Consider the Little Prince of Antoine de Saint-Exupery, a little boywho lives on an extremely small asteroid. Would not the little prince perceiveonly spherical geometry? He would certainly find that a straight line alwaysmeets itself, that a triangle can have more than 2π radians of interior anglemeasure, and the concept of parallelism would have never occurred to him.

Kant assumes geometry is a priori. But if man can impose two differentperceptions of space on reality, each resulting in different judgments, thengeometry must be a posteriori. To use Bostock’s example, there may well also bemen who conceive "7 + 5 = many." Then arithmetical truths are also a posteriori.Now though, we have exhausted Kant’s list of examples, and he seems to haveno proof that there exists mathematical knowledge that is synthetic a priori.

I am not sure as to sufficiency, but I know that Kant considers universalityor non-contingency a necessary quality for a priori truths. Above, I showed howhis examples are really contingent, so they must be synthetic a posteriori. Now,I believe that Kant can attempt to rebut in two ways. The first is that he canencapsulate the perception which the person imposes into the judgment. Thissurely makes the judgment a priori. However, now the predicate is part of thesubject. To refer to an earlier example, now the judgment "7 + 5 = many" isencapsulated as part of the imposed perception. Then we are again at analytica priori, which will not do for Kant. The other rebuttal is to claim that theresults which are invariant under these three geometries are synthetic a prioriand mathematics, but that the others are a posteriori and not mathematics. Butsuch a claim would be, at least to me, a "no true Scotsman" argument.

Part 3

Prompt

In light of your answers to Parts One and Two, please explain how youthink we should understand the ontology, epistemology, and applicabilityof mathematics. (An acceptable answer here is that you have no idea howwe should understand these things. But if this is your answer, you mustexplain why you have no idea!) This part can correctly be addressed inapproximately 500-800 words.

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I personally subscribe to the belief that there are no real, unique mathematicalobjects. For example, I would say that "2" is a symbol, which interacts with othersymbols using predefined rules. Assigning interpretation to "2" is pointless.In this way, I view mathematics as a kind of game. You begin with a set ofsymbols, a set of rules as to how those symbols interact, and possibly rules forhow to generate new symbols. A statement is true with respect to the game ifthe starting symbols can be manipulated, using the rules, to form the statement.So really mathematics is the study of these systems, or the playing of thesegames.

"Is statement xyz true?" is a malformed question. To me, the truth of anystatement is dependent on the system in which it is set. Is this disastrous withrespect to the certainty which Kline seems to value so dearly? It most assuredlyis, but I would claim that "certainty" is irrelevant. There is a certain kind ofaesthetic beauty and pride associated with playing the game, and besides, ifone encapsulates the whole system when uttering a proposition, then one hascertainty anyway (the so called if-then-ism or paraphrase nominalism). By theformer point what I mean to say is that, for example, in the ZFC game I believethat one can create both statements and series of manipulations that are of realbeauty and elegance. This alone makes the game worth playing. Elsewise oneembraces the thesis that any art as a whole is a pointless pursuit. Moreover,playing these games is an interesting pursuit and diversion in and of itself. Ifthe game happens to be played in a system that models the physical well, thenthat is a happy coincidence for the physicists to savour. That however, is noreason to play only in those systems.

I do not think that applicability is the ultimate end which justifies mathe-matics. In this way, I suppose I have circumvented all three of the questions ofapplicability, ontology, and epistemology with rather shallow answers. But thesequestions are really philosophical questions. I believe that I have given a viewthat is deep with respect to mathematics and metamathematics, in that it allowsthe largest number of systems to be examined using mathematical thought. Onepossible objection is that my view sounds a lot like logicism. However, thethesis of logicism is that all mathematics can be reduced to propositions basedon the self-evident axioms of logic. I, however, believe that axioms are whatyou make of them. Their measure of self-evidency only adds to the beauty of aproof for an non-obvious statement, but self-evidency is not necessary in and ofitself. Moreover, I do not believe that logic has any relationship to these symbolgames except inasmuch as many of the games incorporate the rules of logic intotheir rules for symbol manipulation.

As a whole, I find my position a more generalized version of Hilbert’sposition, in that I do accept the rules of the game as arbitrarily stipulatedconstructions. Practically, I do not keep these concepts in the front of my headas I do mathematics. In that way, I suppose my position is at least one partapathy. I am not convinced that this is a bad thing, though; it allows me toremain busy and open-minded. Indeed, I think the phrase, "That’s absurd,you cannot possibly consider this" to be the biggest taboo in mathematics.Empirically, one can observe that the constructivists and finitists are all butfaded away from the mainstream. I am altogether unsure where this position

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lies in the realm of philosophy of mathematics, but it is the only one that I haveconsidered that I find satisfactory.

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